Hi, thanks for your question. My name's XXXXX XXXXX I'm going to assist you with it.
A quote, unlike an estimate, is a fixed priced offer which, when accepted, is legally binding. She should have completed the job for the price quoted, even if she did miscalculate the amount of work involved. If, when she texted you, you simply told her to carry on, then you have done nothing other than confirm this. Even if the nature of your text implied that you might pay her an additional fee, this will not usually be deemed valid "consideration" (i.e. money she is contractually entitled to claim from you) as it relates to a contractual duty she already owed you.
She may argue that the £175 was only an estimate but the fact that you paid her in full upfront (rather than just giving her a deposit) would suggest otherwise. Further, where an estimate is given, she is not permitted to view this as a "blank cheque" and any variation above this should be reasonable. From the information provided in your question, at no time did she make you aware that the additional work required was so excessive that it would triple the original amount.
If you want to nip this in the bud, you could try and negotiate with her, on a without prejudice basis, and perhaps offer to pay her a sum you are comfortable with (say, the £40 you envisaged), pointing out that the original quote was binding but it is simply a goodwill gesture to dispose of the matter.
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